$f(x, y) = (y + \sin(y), x - \cos(x))$ $\text{curl}(f) = $
Answer: The formula for curl in two dimensions is $\text{curl}(f) = \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}$, where $P$ is the $x$ -component of $f$ and $Q$ is the $y$ -component. Let's differentiate! $\begin{aligned} \dfrac{\partial Q}{\partial x} &= \dfrac{\partial}{\partial x} \left[ x - \cos(x) \right] \\ \\ &= 1 + \sin(x) \\ \\ \dfrac{\partial P}{\partial y} &= \dfrac{\partial}{\partial y} \left[ y + \sin(y) \right] \\ \\ &= 1 + \cos(y) \end{aligned}$ Therefore: $\begin{aligned} \text{curl}(f) &= 1 + \sin(x) - (1 + \cos(y)) \\ \\ &= \sin(x) - \cos(y) \end{aligned}$